Exact Hamiltonian Research Articles

Ab Initio Computation of Auger Decay in Heavy Metals: Zinc about It.

We report the first coupled-cluster study of Auger decay in heavy metals. The zinc atom is used as a case study due to its relevance to the Auger emission properties of the 67Ga radionuclide. Coupled-cluster theory combined with complex basis functions is used to describe the transient nature of the core-ionized zinc atom. We also introduce second-order Møller-Plesset perturbation theory as an alternative method for computing partial Auger decay widths. Scalar-relativistic effects are included in our approach for computing Auger electron energies by means of the spin-free exact two-component one-electron Hamiltonian, while spin-orbit coupling is treated by means of perturbation theory. We center our attention on the K-edge Auger decay of zinc dividing the spectrum into three parts (K-LL, K-LM, and K-MM) according to the shells involved in the decay. The computed Auger spectra are in good agreement with experimental results. The most intense peak is found at an Auger electron energy of 7432 eV, which corresponds to a 1D2 final state arising from K-L2L3 transitions. Our results highlight the importance of relativistic effects for describing Auger decay in heavier nuclei. Furthermore, the effect of a first solvation shell is studied by modeling Auger decay in the hexaaqua-zinc(II) complex. We find that K-edge Auger decay is slightly enhanced by the presence of the water molecules as compared to the bare atom.

Shadow Hamiltonians of structure-preserving integrators for Nambu mechanics

Abstract Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is a kind of generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is nontrivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is nontrivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In this paper we present a general procedure to calculate the shadow Hamiltonians of structure-preserving integrators for Nambu mechanics, and give an example where the shadow Hamiltonians exist. This is the first attempt to determine the concrete forms of the shadow Hamiltonians for a Nambu mechanical system. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role in calculating the shadow Hamiltonians using the Baker–Campbell–Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

Foldy-Wouthuysen transformation and multiwave states of a graphene electron in external fields and free (2+1)-space

The relativistic Foldy-Wouthuysen transformation is used for an advanced description of planar graphene electrons in external fields and free (2+1)-space. It is shown that the initial Dirac equation should by based on the usual (4 × 4) Dirac matrices but not on the reduction of matrix dimensions and the use of (2 × 2) Pauli matrices. Nevertheless, the both approaches agree with the experimental data on graphene electrons in a uniform magnetic field. The pseudospin of graphene electrons is not the one-value spin and takes the values ±1/2. The exact Foldy-Wouthuysen Hamiltonian of a graphene electron in uniform and nonuniform magnetic fields is derived. The exact energy spectrum agreeing with the experiment and exact Foldy-Wouthuysen wave eigenfunctions are obtained. These eigenfunctions describe multiwave (structured) states in the (2+1)-space. It is proven that the Hermite-Gauss beams exist even in the free space. In the multiwave Hermite-Gauss states, graphene electrons acquire nonzero effective masses dependent on a quantum number and move with group velocities which are less than the Fermi velocity. Graphene electrons in a static electric field also can exist in the multiwave Hermite-Gauss states defining non-spreading coherent beams. These beams can be accelerated and decelerated.

Electron correlation effects on uranium isotope fractionation in U(VI)-U(VI) and U(IV)-U(VI) equilibrium isotopic exchange systems.

Uranium isotope fractionation has been extensively investigated in the fields of nuclear engineering and geochemical studies, yet the underlying mechanisms remain unclear. This study assessed isotope fractionations in U(VI)-U(VI) and U(IV)-U(VI) systems by employing various relativistic electron correlation methods to explore the effect of electron correlation and to realize accurate calculations of isotope fractionation coefficients (ε). The nuclear volume term, ln Knv, the major term in ε, was estimated using the exact two-component relativistic Hamiltonian in conjunction with either HF, DFT(B3LYP), MP2, CCSD, CCSD(T), FSCCSD, CASPT2, or RASPT2 approaches for small molecular models with high symmetry. In contrast, chemical species studied in prior experimental work had moderate sizes and were asymmetrical. In such cases, electron correlation calculations other than DFT calculations were not possible and so only the HF and B3LYP approaches were employed. For closed-shell U(VI)-U(VI) systems, the MP2, CCSD and CCSD(T) methods yielded similar ln Knv values that were intermediate between those for the HF and B3LYP methods. Comparisons with experimental results for U(VI)-U(VI) systems showed that the B3LYP calculations gave results closer to the experimental data than the HF calculations. Because of the open-shell structure of U(IV), multireference methods involving the FSCCSD, CASPT2 and RASPT2 techniques were used for U(IV)-U(VI) systems, but these calculations exhibited instability. The average-of-configuration HF method showed better agreement with the experimental ε values for U(IV)-U(VI) systems than the B3LYP method. Overall, electron correlation improved the description of ε for the U(VI)-U(VI) systems but challenges remain with regard to open-shell U(IV) calculations because an energy accuracy of 10-6-10-7Eh is required for ln Knv calculations.

Guiding centre motion for particles in a ponderomotive magnetostatic end plug

The Hamiltonian dynamics of a single particle in a rotating plasma column, interacting with an magnetic multipole is perturbatively solved for up to second order, using the method of Lie transformations. First, the exact Hamiltonian is expressed in terms of canonical action-angle variables, and then an approximate integrable Hamiltonian is introduced, using another set of actions and angles, which describe the centre of oscillation for the particle. The perturbation introduces an effective ponderomotive potential, which to leading order is positive. At the second order, the pseudopotential consists of a sum of terms of the Miller form, and can have either sign. Additionally, at second order, the ponderomotive interaction introduces a modification to the particle effective mass, when considering the motion along the column axis. It is found that particles can be axially confined by the ponderomotive potentials, but acquire radial excursions which scale as the confining potential. The radial excursions of the particle along its trajectory are investigated, and a condition for the minimal rotation frequency for which the particle remains radially confined is derived. Last, we comment on the changes to the aforementioned solution to the pseudopotentials and particle trajectory in the case of resonant motion, that is, a motion which has the same periodicity as the perturbation.

SOiCISCF: Combining SOiCI and iCISCF for Variational Treatment of Spin-Orbit Coupling.

It has recently been shown that the SOiCI approach [Zhang, N.; J. Phys.: Condens. Matter 2022, 34, 224007], in conjunction with the spin-separated exact two-component relativistic Hamiltonian, can provide very accurate fine structures of systems containing heavy elements by treating electron correlation and spin-orbit coupling (SOC) on an equal footing. Nonetheless, orbital relaxations/polarizations induced by SOC are not yet fully accounted for due to the use of scalar relativistic orbitals. This issue can be resolved by further optimizing the still real-valued orbitals self-consistently in the presence of SOC, as done in the spin-orbit coupled CASSCF approach [Ganyushin, D.; et al. J. Chem. Phys. 2013, 138, 104113] but with the iCISCF algorithm [Guo, Y.; J. Chem. Theory Comput. 2021, 17, 7545-7561] for large active spaces. The resulting SOiCISCF employs both double group and time reversal symmetries for computational efficiency and the assignment of target states. The fine structures of p-block elements are taken as showcases to reveal the efficacy of SOiCISCF.

Comparison of state-interaction and spinor-representation calculations of spin-orbit coupling within exact two-component coupled-cluster theories

A benchmark study of state-interaction and spinor-representation calculations of spin-orbit coupling using the exact two-component Hamiltonians with atomic mean-field integrals (the X2CAMF schemes) at the equation-of-motion coupled-cluster singles and doubles level is reported. We adopt a version of the X2CAMF scheme with spin-orbit integrals correct to first order for the state-interaction calculations and a version correct to infinite order for the spinor-representation calculations. The differences between the state-interaction calculations with minimum active spaces to account for (quasi-)degeneracy and the spinor-representation calculations thus correspond to higher-order spin-orbit contributions. This state-interaction approach with scalar-relativistic effects accurately included in the reference functions and in the spin-orbit integrals is shown to exhibit robust performance for elements across the periodic table. On the other hand, the more rigorous spinor representation shows more rapid convergence with respect to the number of correlated electrons and is the preferred choice for accurate calculations for heavy elements.

Approximation of the ground and excited states of atoms and molecules with a single determinant by means of the exact many-particle Hamiltonian

In the present work, we search for the maximum of the functional | ⟨ Φ ′ | H ^ | Φ ⟩ | , where | Φ ′ ⟩ is a Slater Determinant (SlDet) and H ^ is the exact Hamiltonian of an atom or a molecule, starting from a SlDet | Φ ⟩ , with its spin orbitals calculated by the standard Hartree-Fock (HF) equation or other approximation or any determinant. The element | ⟨ Φ 1 | H ^ | Φ ⟩ | with | Φ 1 ⟩ the maximising | Φ ′ ⟩ gives a value greater or equal to | ⟨ Φ | H ^ | Φ ⟩ | . Next we calculate the corresponding maximum overlap | ⟨ Φ 2 | H ^ | Φ 1 ⟩ | and finally | ⟨ Φ n + 1 | H ^ | Φ n ⟩ | until | ⟨ Φ n + 1 | H ^ | Φ n ⟩ − ⟨ Φ n | H ^ | Φ n − 1 ⟩ | ≤ ε , where ε determines the desired numerical accuracy. We show that the sequence ⟨ Φ n + 1 | H ^ | Φ n ⟩ converges to an extremum ⟨ Φ | H ^ | Φ ⟩ . Having found the SlDet with the lowest energy we repeat the procedure in the orthogonal subspace for finding the first and the higher excited states. We applied this method in order to determine the energies of several configurations of H 3 , the Lithium atom, LiH and Be.

Interaction mediated electron transport within the many-body tunnelling Hamiltonian

For a non-superconducting system and in the spirit of the transfer or tunnelling Hamiltonian formalism, an expression for the electronic tunnelling current through an insulating barrier is calculated, explicitly taking into account the effects of electron-electron interactions that can persist across the barrier. In the presence of a single tunnelling barrier, the exact Hamiltonian is projected onto the subspaces of the ‘left’ and ‘right’ conducting leads. In the weak-tunnelling limit, the well-known tunnelling Hamiltonian is recovered, along with an additional term. This additional term originates from the projection of the electron-electron interaction onto each subspace and corresponds to a correlated or interaction-mediated tunnelling. It is shown that the tunnelling current in this approximation is related to—in addition to the single-particle density of states—the two-particle spin-spin and density-density susceptibilities. The signatures of these terms have been observed in scanning tunnelling microscope experiments on atomic spin chains.

On the Bipolaronic Mechanism of High-Temperature Superconductivity in "Ginzburg Sandwiches" FeSe-SrTiO<sub>3</sub>; SrTiO<sub>3</sub>-FeSe-SrTiO<sub>3</sub>

The mechanism of experimentally observed high-temperature superconductivity (HTSC) in thin FeSe films on SrTiO₃-type substrates has been theoretically investigated. Applying the theory of large-radius bipolaronic states developed based on the exact Hamiltonian of electron-phonon interaction for arbitrary multilayer structures, the bipolaronic mechanism of Cooper pairing of polarons in FeSe monolayers on SrTiO₃ substrates is investigated and in three-layer structures SrTiO₃–FeSe–SrTiO₃, which are typical "Ginzburg sandwiches". Approach proposed by Ginzburg to enhance the electron-phonon interaction and achieve HTSС by separating the regions where electrons are located (forming Cooper pairs or bipolarons) with the regions in which excitons are excited (or inertial polarization is induced), made it possible to implement the criteria for the formation of bipolaronic states in multilayer structures with high binding energy, due to the possibility of selecting optimal geometric and material parameters (layer thicknesses, dielectric permittivity, optical frequencies, effective masses). It is shown that the binding energy of bipolarons (<i>E_bp</i>) in these structures is in the range of values for which bipolarons remain stable quasiparticles and can exist at temperatures significantly higher than their Bose condensation temperature. The formation of bipolarons with high binding energy in the FeSe monolayer on the SrTiO₃ substrate provides the emergence of a bipolaronic HTS with a critical temperature (<i>T_c</i>) more than an order of magnitude higher than <i>T_c</i> for massive FeSe crystals. At the same time, the binding energy of the bipolaron in the FeSe layer with thickness <i>d</i> on the SrTiO₃ substrate increases exponentially with decreasing <i>d</i> (<i>E_bp</i>~<i>exp</i>(-<i>d </i>/ <I>R<SUB>S</SUB></I>) <I>R<SUB>S</SUB></I> is the radius of the polaron) and reaches its maximum value in the limit of the multilayer film FeSe (d→0). The presented theory allows modeling a multilayer system and determining the range of values of the material and geometric parameters of layers forming a multilayer structure with a large number of FeSe layers in which <i>T_c</i> values in the room temperature range can be achieved.

Nuclear Structure Investigations of Even–Even Hf Isotopes

The mass region of rare-earth nuclei in the nuclear chart is riddled with well-deformed nuclei, exhibiting rotational properties and many interesting nuclear structure-related phenomena. The scarcity of experimental data as the neutron number increases and the exotic phenomena such as shape coexistence, which are strongly connected with the underlying symmetries of the Hamiltonian and are predicted to take place in this region, make this mass region a fertile ground for experimental and theoretical studies of nuclear structure. In this work, we investigate the structure of the even–even 162–184Hf (hafnium) isotopes through a calculation of various observables such as B(E2;01+→21+) reduced transition matrix elements and quadrupole moments. Six different nuclear models are employed in the calculations of the observables for these nuclei, the shapes of which deviate from spherical symmetry, and as such, are characterized by Hamiltonians, which break the rotational invariance of the exact nuclear many-body Hamiltonian. The results of the present study are expected to establish some concrete guidelines for current and future experimental endeavors. Along these lines, the results for the 162–180Hf isotopes are compared with existing experimental data where available, showing an overall good agreement.

Composite fermion theory: A microscopic derivation without Landau level projection

The composite fermion (CF) theory gives both a phenomenological description for many fractional quantum Hall (FQH) states, as well as a microscopic construction for large scale numerical calculation of these topological phases. The fundamental postulate of mapping FQH states of electrons to integer quantum Hall (IQH) states of CFs, however, was not formally established. The Landau level (LL) projection needed for the microscopic calculations is in some sense uncontrolled and unpredictable. We rigorously derive the unitary relationship between electrons and the CFs, showing the latter naturally emerge from special interactions within a single LL, without resorting to any projection by hand. In this framework, all FQH states topologically equivalent to those described by the conventional CF theory (e.g., the Jain series) have exact model Hamiltonians that can be explicitly derived, and we can easily generalize to FQH states from interacting CFs. Our derivations reveal fundamental connections between the CF theory and the pseudopotential/Jack polynomial constructions, and argue that all Abelian CF states are physically equivalent to the IQH states, while a plethora of non-Abelian CF states can be systematically constructed and classified. We also discuss about implications to experiments and effective field theory descriptions based on the descriptions with CFs as elementary particles.

On the conservative character of discretizations to Itô-Hamiltonian systems with small noise

In this paper, we consider stochastic Hamiltonian systems of Itô type driven by a multiplicative small noise. It is well-known, indeed, that stochastic Hamiltonian problems are suitable candidates to reconcile the Hamiltonian classical mechanics with the non-differentiability of the Wiener process, which provides the innovative character of stochastic effects. In particular, in this work, we provide a characterization of the behavior of averaged Hamiltonians arised in such systems, with more emphasis on the separable and quadratic Hamiltonians. Next, we show that, in general, first order approximations to such systems are not able to retain the same behavior discovered for the exact averaged Hamiltonian. Hence, the analysis for the specific case of ϑ-methods is performed. Finally, numerical evidence is depicted in order to confirm theoretical results.

Quantum algorithms for Schrieffer-Wolff transformation

The Schrieffer-Wolff transformation aims to solve degenerate perturbation problems and give an effective Hamiltonian that describes the low-energy dynamics of the exact Hamiltonian in the low-energy subspace of unperturbed Hamiltonian. This unitary transformation decoupling the low-energy and high-energy subspaces for the transformed Hamiltonian can be realized by quantum circuits. We give a fully quantum algorithm for realizing the SW transformation. We also propose a hybrid quantum-classical algorithm for finding the effective Hamiltonian on NISQ hardware, where a general cost function is used to indicate the decoupling degree. Numerical simulations without or with noise and experiments on quantum computer ibmq_manila are implemented for a Heisenberg chain model. Our results verify the algorithm and show its validity on near-term quantum computers.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple.

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnFn, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Non-Abelian chiral spin liquid on a spin-1 kagome lattice: Truncation of an exact Hamiltonian and numerical optimization

We search for short-range Hamiltonians of finite spin-1 kagome systems, maximizing the overlaps with lattice Moore-Read states. Our starting point is an exact, long-range parent Hamiltonian for such a state on a finite plane, obtained from conformal field theory. A truncation procedure is applied to it, which retains only short-range terms and makes it easy to define the Hamiltonian on a torus. Finally, the remaining coefficients are optimized, to yield maximized overlaps between exact diagonalization results and model ground states. In the best cases, these overlaps exceed 0.9 and 0.8 for the three lowest states of 12- and 18-site systems, respectively, suggesting that the obtained Hamiltonians are good parent Hamiltonians for a non-Abelian topological order.

A (semi)-exact Hamiltonian for the curvature perturbation ζ

The total Hamiltonian in general relativity, which involves the first class Hamiltonian and momentum constraints, weakly vanishes. However, when the action is expanded around a classical solution as in the case of a single scalar field inflationary model, there appears a non-vanishing Hamiltonian and additional first class constraints; but this time the theory becomes perturbative in the number of fluctuation fields. We show that one can reorganize this expansion and solve the Hamiltonian constraint exactly, which yield an explicit all order action. On the other hand, the momentum constraint can be solved perturbatively in the tensor modes γ ij by still keeping the curvature perturbation ζ dependence exact. In this way, after gauge fixing, one can obtain a semi-exact Hamiltonian for ζ which only gets corrections from the interactions with the tensor modes (hence the Hamiltonian becomes exact when the tensor perturbations set to zero). The equations of motion clearly exhibit when the evolution of ζ involves a logarithmic time dependence, which is a subtle point that has been debated in the literature. We discuss the long wavelength and late time limits, and obtain some simple but non-trivial classical solutions of the ζ zero-mode.

Relativistic calculations on the transition electric dipole moments and radiative lifetimes of the spin-forbidden transitions in the antimony hydride molecule

Calculations on the spectroscopic constants and transition properties of the first three states (a1Δ, b1Σ+, and X3Σ−) of the SbH molecule were performed under the relativistic framework using the exact two-component Hamiltonian (X2C). The potential energy curves in the Franck–Condon region were computed and compared with the previous values. Furthermore, the transition dipole moments for the weak spin-forbidden transitions (b0+–X10+, b0+–X21, X10+–X21, and X21–a2) were reported. The spontaneous radiative lifetime of the b1Σ+ (υ′ = 0) state was calculated as 163.5±7.5 μ s, which is in reasonable agreement with the latest experimental value of 173±3 μ s. The spontaneous radiative lifetimes of the X21 (υ′ = 0) state and the a2 (υ′ = 0) state were calculated to be 48.6 s and ∼ 8 ms, respectively. Our study is expected to be a benchmark transition property computation for comparison with other theoretical and experimental results. The datasets presented in this paper, including the transition dipole moments, are openly available at https://dx.doi.org/10.11922/sciencedb.j00113.00018.

Enhancement of the parity-violating energy difference of H2X2 molecules by electronic excitation

The parity-violating energy difference (PVED) between two enantiomers of a chiral molecule is caused by the weak interaction. Because of the smallness of the PVED, nonzero PVED is yet to be discovered in experimental searches. To detect the PVED, the search for molecules with large PVED values is important. Previously, one of the authors proposed that the PVED may be significantly enhanced in ionized or excited states. The significant enhancement of the PVED in some electronic excited states is proven in this study using H$_2$X$_2$ (X=O, S, Se, Te) molecules as examples. The maximum enhancement was an about 360-fold increase for H$_2$Se$_2$. For the PVED calculation, we employ the finite-field perturbation theory (FFPT) within the equation-of-motion coupled-cluster theory based on the exact two-component molecular-mean field Hamiltonian. The relation between the enhancement of the PVED and the contribution to the PVED from the highest occupied molecular orbital is also examined. The effects of computational elements, such as parameters related to the electron correlation and FFPT on PVED values in excited states of H$_2$X$_2$ molecules are studied.

Perturbation-adapted perturbation theory.

A new general approach is introduced for defining an optimum zero-order Hamiltonian for Rayleigh-Schrödinger perturbation theory. Instead of taking the operator directly from a model problem, it is constructed to be a best fit to the exact Hamiltonian within any desired functional form. When applied to many-body perturbation theory for electrons, strongly improved convergence is observed in cases where the conventional Fock Hamiltonian leads to divergence or slow convergence.